The New Zealanders ran in seven tries, all but two in the first half, with centres Matt Proctor and Vince Aso and prop Reggie Goodes also grabbing scores to go with 15 points from the boot of flyhalf Beauden Barrett.
The Lions were never in the contest, but still managed three tries through hooker Armand van der Merwe, centre Lionel Mapoe and flanker Jaco Kriel.
The visitors raced into a 21-0 lead with just a quarter of the game gone as their slick handling proved too much for the home side.
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Savea crossed the line in the corner after some excellent build-up play, though there was a suspicion of a knock-on before the ball was spun wide to the All Black wing.
Proctor raced away to score through an interception before Jane grabbed his first try in controversial fashion when he strolled over the line as referee Marius van der Westhuizen unwittingly obstructed the defenders, yet allowed the try to stand.
The Lions briefly rallied as Van der Merwe scored a try from a powerful maul but that joy, and any realistic thoughts of a comeback, were shortlived as Jane got the Hurricanes' second intercept score of the game.
And Goodes capped off a remarkable first half for the visitors with a fine solo effort as he bashed through defenders close to the touchline and managed to dot down to give the Wellington-based side a 36-5 lead at the break.
The visitors started the second half on the front foot but it was the Lions who scored first when wing Ruan Combrinck set up an easy run-in for Mapoe.
The home side began to dominate territory and got their third score when Kriel barged his way over in the corner, but a turnover then allowed the visitors to break away and restore their bonus point as Aso cantered in.
Savea added the final try with a burst down the touchline to cap a sixth win of the season for the Hurricanes that leaves them in fourth in the Australasian Group. The Lions stay five points clear in the Africa Conference 2.
(Reporting By Nick Said, editing by Ian Chadband)
